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Alternating distances of knots and links
Topology and its Applications 182 (2015) 53–70
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Topology and its Applications www.elsevier.com/locate/topol
Alternating distances of knots and links Adam M. Lowrance 1 Department of Mathematics, Vassar College, Poughkeepsie, NY, United States
a r t i c l e
i n f o
Article history: Received 26 June 2014 Accepted 7 December 2014 Available online xxxx Keywords: Alternating knot Dealternating Almost alternating Alternation Turaev genus Alternating genus Toroidally alternating Jones polynomial Warping polynomial Warping degree Knot theory Khovanov homology Knot Floer homology Homological width
a b s t r a c t An alternating distance is a link invariant that measures how far away a link is from alternating. We study several alternating distances and demonstrate that there exist families of links for which the difference between certain alternating distances is arbitrarily large. We also show that two alternating distances, the alternation number and the alternating genus, are not comparable. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Alternating links play an important role in knot theory and 3-manifold geometry and topology. Link invariants are often easier to compute and take on special forms for alternating links. Moreover, the complements of alternating links have interesting topological and geometric structures. Many generalizations of alternating links exist, and a particular generalization can give rise to an invariant that measures how far a link is from alternating. We study several such invariants, which we call alternating distances. A link L is split if it has a separating sphere, i.e. a two-sphere S 2 in S 3 such that L and S 2 are disjoint and each component of S 3 − S 2 contains at least one component of L. We will mostly be concerned with non-split links, that is links with no separating spheres. A real valued link invariant d(L) is an alternating distance if it satisfies the following conditions.
1
E-mail address: [email protected]. The author was supported by an AMS–Simons Travel Grant.
http://dx.doi.org/10.1016/j.topol.2014.12.010 0166-8641/© 2014 Elsevier B.V. All rights reserved.
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(1) For any non-split link, d(L) ≥ 0. (2) For any non-split link, d(L) = 0 if and only if L is alternating. (3) If L1 and L2 are non-split links and L1 #L2 is any connected sum of L1 and L2 , then d(L1 #L2 ) ≤ d(L1 ) + d(L2 ). The connected sum L1 #L2 depends on a choice of components in L1 and L2 . However, condition (3) above must be true for any choice of connected sum. We will frequently use the notation L1#L2 to denote an arbitrary choice of connected sum. We consider the following invariants. The dealternating number, denoted by dalt(L), and the alternation number, denoted by alt(L), are defined by counting crossing changes. The relationship between the minimum crossing number c(L) of a link L and the span of the Jones polynomial VL (t) of L was used to prove some of Tait’s famous conjectures, and we study the difference c(L) − span VL (t). The Turaev genus, denoted by gT (L), and the alternating genus, denoted by galt (L), are the genera of certain surfaces associated to L. The warping span, denoted by warp(K), is defined by examining the over-under behavior as one travels along the knot or link. Precise definitions of these invariants are given in Section 2. Let d1 and d2 be real valued link invariants and let F be a family of links. We say d2 dominates d1 on F, and write d1 (F) d2 (F), if for each positive integer n, there exists a link Ln ∈ F such that d2 (Ln ) − d1 (Ln ) ≥ n. In Section 4, we examine three families of links. The first family F(Wn ) consists of iterated Whitehead doubles of the figure-eight knot. The second family F(T(p, q)) consists of links obtained by changing certain crossings of torus links. The third family F(T (3, q)) consists of the (3, q)-torus knots. Theorem 1.1. Let F(Wn ), F(T(p, q)), and F(T (3, q)) be the families of links above. (1) The dealternating number, c(L) − span VL (t), and Turaev genus dominate the alternation number on F(Wn ). (2) The dealternating number, Turaev genus, c(L) − span VL (t), and alternation number dominate the alternating genus on F(T(p, q)). (3) The dealternating number, alternation number, c(L) −span VL (t), and Turaev genus dominate the warping span on F(T (3, q)). (4) The difference c(L) − span VL (t) dominates the dealternating number, alternation number, alternating genus, and Turaev genus on F(T (3, q)). Two real valued link invariants d1 and d2 are said to be comparable if either d1 (L) ≤ d2 (L) or d2 (L) ≤ d1 (L) for all links L. The invariants d1 and d2 are not comparable if there exist links L and L such that d1 (L) < d2 (L) and d1 (L ) > d2 (L ). Theorem 1.2. The alternation number and alternating genus of a link are not comparable. This paper is organized as follows. In Section 2, we define the invariants mentioned in Theorems 1.1 and 1.2. In Section 3, we describe some lower bounds for the invariants. In Section 4, we define several families of links and use them to prove Theorems 1.1 and 1.2. In Section 5, we discuss some open questions about our invariants. 2. The invariants In this section, the invariants of Theorems 1.1 and 1.2 are defined. We show that each one is an alternating distance, and discuss some known relationships between them.
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2.1. Dealternating number Adams et al. [3] define almost alternating links to be non-alternating links with a diagram such that one crossing change makes the diagram alternating. They use the notion of number of crossing changes needed to make a diagram alternating to define the dealternating number of a link. The dealternating number of a link diagram D, denoted by dalt(D), is the minimum number of crossing changes necessary to transform D into an alternating diagram. The dealternating number of a link L, denoted by dalt(L), is the minimum dealternating number of any diagram of L. A link with dealternating number k is also called k-almost alternating. Proposition 2.1. The dealternating number of a link is an alternating distance. Proof. The definition of the dealternating number implies that it is always non-negative and equals zero if and only if the link is alternating. Suppose that L1 and L2 are links with diagrams D1 and D2 such that dalt(D1 ) = dalt(L1 ) and dalt(D2 ) = dalt(L2 ). For any choice of connected sum L1 #L2 , there exists some choice D1 #D2 of connected sum of diagrams D1 and D2 such that D1 #D2 is a diagram of L1 #L2 and dalt(D1 #D2 ) = dalt(D1 ) + dalt(D2 ) = dalt(L1 ) + dalt(L2 ). Thus dalt(L1 #L2 ) ≤ dalt(L1 ) + dalt(L2 ) and hence the dealternating number of a link is an alternating distance. 2 2.2. Alternation number Kawauchi [20] uses crossing changes in a slightly different manner to define the alternation number of a link. The alternation number of a link diagram D, denoted by alt(D), is the minimum number of crossing changes necessary to transform D into some (possibly non-alternating) diagram of an alternating link. The alternation number of a link L, denoted by alt(L), is the minimum alternation number of any diagram of L. The alternation number of L is also the Gordian distance from L to the set of alternating links [28]. It is immediate from their definitions that alt(L) ≤ dalt(L)
(2.1)
for any link L. Proposition 2.2. The alternation number of a link is an alternating distance. Proof. The definition of the alternation number implies that it is always non-negative and equals zero if and only if the link is alternating. Suppose that L1 and L2 are links with diagrams D1 and D2 respectively such 1 and D 2 be the diagrams of alternating links obtained that alt(D1 ) = alt(L1 ) and alt(D2 ) = alt(L2 ). Let D from D1 and D2 respectively via the minimum number of crossing changes. Let L1 #L2 be a connected sum of L1 and L2 , and let D1 #D2 be a connected sum of D1 and D2 such that D1 #D2 is a diagram of L1 #L2 . 2 (an alternating link) via alt(D1 ) +alt(D2 ) = alt(L1 ) +alt(L2 ) 1 #D Then D1 #D2 can be transformed into D crossing changes. Hence alt(L1 #L2 ) ≤ alt(L1 ) + alt(L2 ), and thus the alternation number is an alternating distance. 2 2.3. Crossing number and the span of the Jones polynomial In the late 19th century, Tait [41] conjectured that a certain type of alternating link diagram (called reduced) has minimal crossing number among all diagrams for that link. This conjecture remained undecided until the discovery of the Jones polynomial [16], when combined work of Kauffman [19] and Murasugi [29] proved the conjecture to be true.
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Fig. 1. The A and B resolutions of a crossing.
Fig. 2. In a neighborhood of each vertex of Γ a saddle surface transitions between the all-A and all-B states.
Let VL (t) denote the Jones polynomial of L, and let max deg VL (t) and min deg VL (t) denote the maximum and minimum power of t in VL (t) with non-zero coefficient respectively. Define span VL (t) = max deg VL (t) − min deg VL (t). Let c(L) denote the minimum number of crossings in any diagram of the link L. Proposition 2.3. The difference c(L) − span VL (t) is an alternating distance. Proof. Murasugi [29] and Kaufman [19] prove that span VL (t) ≤ c(L) for any non-split link L. Furthermore, when L is non-split, Murasugi proves that span VL (t) = c(L) if and only if L is a connected sum of alternating links, which happens if and only if L is alternating. The span of the Jones polynomial is additive under connected sums, i.e. span VL1 #L2 (t) = span VL1 (t) +span VL2 (t). It is a long-standing open question whether crossing number is additive, but it is easy to see that crossing number is sub-additive, i.e. c(L1 #L2 ) ≤ c(L1 ) + c(L2 ). Therefore c(L1 #L2 ) − span VL1 #L2 (t) ≤ (c(L1 ) − span VL1 (t)) + (c(L2 ) − span VL2 (t)). Hence c(L) − span VL (t) is an alternating distance. 2 2.4. Turaev genus Turaev [43] gives a simplified proof of Tait’s conjecture mentioned above, where he associates to each link diagram D a surface F (D), now known as the Turaev surface of D. Each crossing of D can be resolved in either an A-resolution or a B-resolution, as depicted in Fig. 1. A collection s of simple closed curves obtained by choosing either an A-resolution or a B-resolution for each crossing of D is a state of D. The state sA (D) obtained by choosing an A-resolution at each crossing is called the all-A state of D, and similarly, the state sB (D) obtained by choosing a B-resolution at each crossing is called the all-B state of D. Let Γ be the 4-valent graph obtained from D by forgetting the “over-under” information at each crossing. Regard Γ as embedded in S 2 , and thicken the sphere to S 2 × [−1, 1] such that Γ is a subset of S 2 × {0}. We first describe the intersection of the Turaev surface F (D) and S 2 × [−1, 1]. Outside of neighborhoods of the vertices of Γ , the Turaev surface intersects S 2 × [−1, 1] in Γ × [−1, 1]. In a neighborhood of each crossing, the Turaev surface intersects S 2 × [−1, 1] in a saddle positioned so that F (D) ∩ S 2 × {−1} is the all-B state sB (D) and F (D) ∩ S 2 × {1} is the all-A state sA (D), as depicted in Fig. 2. Summarizing, the intersection of the Turaev surface with S 2 × [−1, 1] is a cobordism between sA (D) and sB (D) whose saddle points correspond to the crossings of D. Outside of S 2 × [−1, 1] the Turaev surface F (D) is a collection of disks capping off the components of sA (D) and sB (D).
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The Turaev surface F (D) is always oriented, and if D is a diagram of a non-split link L, then F (D) is connected. The Turaev genus gT (D) of a diagram D is defined to be the genus of the Turaev surface F (D). It can be shown that gT (D) =
1 2 + c(D) − sA (D) − sB (D) , 2
(2.2)
where c(D) is the number of crossings of D, and |sA (D)| and |sB (D)| denote the number of components in the all-A and all-B states of D. The Turaev genus gT (L) of a non-split link L is the minimum genus of the Turaev surface of D where D is any diagram of L, i.e. gT (L) = min gT (D) D is a diagram of L . A closed, oriented surface Σ ⊂ S 3 is a Heegaard surface if both components of S 3 − Σ are handlebodies. If Σ is a Heegaard surface in S 3 , then every link L has an isotopy class representative that lies within a thickened neighborhood Σ × [−1, 1] of Σ. Let π : Σ × [−1, 1] → Σ be projection onto the first factor. The link L is said to have an alternating projection to Σ if π(L) has only transverse double points and the crossings of the projection π(L) alternate as one travels along each component of L. Dasbach et al. [11] prove the following facts about the Turaev surface F (D) and Turaev genus of a non-split link. • • • •
The The The The
Turaev surface F (D) is a Heegaard surface in S 3 . link L has an alternating projection π to F (D). complement F (D) − π(L) of the projection π is a disjoint union of disks. Turaev surface of an alternating diagram is a sphere, and gT (L) = 0 if and only if L is alternating.
Proposition 2.4. The Turaev genus of a link is an alternating distance. Proof. Since Turaev genus is a minimum genus of a surface, it is always non-negative. As mentioned above, a link L has Turaev genus zero if and only if it is alternating. If D1 and D2 are link diagrams and D1 #D2 is any connected sum, then |sA (D1 #D2 )| = |sA (D1 )| + |sA (D2 )| − 1 and |sB (D1 #D2 )| = |sB (D1 )| + |sB (D2 )| − 1. Then Eq. (2.2) implies that gT (D1 #D2 ) = gT (D1 ) +gT (D2 ). Hence if L1 and L2 are links and L1 #L2 is any connected sum, then gT (L1 #L2 ) ≤ gT (L1 ) + gT (L2 ). Therefore, the Turaev genus of a link is an alternating distance. 2 Turaev [43] shows that gT (D) ≤ c(D) − span VL (t) for any diagram D of the link L. Minimizing over all diagrams of the link L, one obtains gT (L) ≤ c(L) − span VL (t).
(2.3)
Abe and Kishimoto [2] examine the behavior of the Turaev surface under crossing changes to show that gT (D) ≤ dalt(D) for any link diagram D. Consequently, for any link L, gT (L) ≤ dalt(L).
(2.4)
Champanerkar and Kofman’s recent survey [8] gives many open questions concerning the Turaev genus of a link.
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2.5. Alternating genus Following his work on almost alternating links, Adams [4] defined toroidally alternating links as those links L that have an alternating projection π to a Heegaard torus Σ such that the complement of the projection in the Heegaard torus, i.e. Σ − π(L), is a disjoint union of disks. Toroidally alternating links can be naturally generalized. Define the alternating genus of a non-split link L, denoted by galt (L), to be the minimum genus of any Heegaard surface Σ such that the link L lies within a regular neighborhood Σ × [−1, 1] of the Heegaard surface, has an alternating projection π : Σ × [−1, 1] → Σ, and Σ − π(L) is a disjoint union of disks. For any diagram D of L, the Turaev surface F (D) is an example of such a surface, and so galt (L) is well-defined, and galt (L) ≤ gT (L)
(2.5)
for any link L. Proposition 2.5. The alternating genus of a link is an alternating distance. Proof. By definition, a non-split link L has alternating genus zero if and only if it has an alternating projection to a sphere, i.e. if and only if L is alternating. Moreover, since alternating genus is the minimum genus of some surface, it is always non-negative. Let L1 #L2 be a connected sum of links L1 and L2 such that both L1 and L2 have an alternating projections π1 and π2 to respective Heegaard surfaces Σ1 and Σ2 . Moreover, suppose that both Σ1 − π1 (L1 ) and Σ2 − π2 (L2 ) are disjoint unions of disks. Then there exist disks D1 and D2 in Σ1 and Σ2 meeting the projections π(L1 ) and π(L2 ) in a single arc such that L1 #L2 has an alternating projection to Σ1 #Σ2 whose complementary regions are a disjoint union of disks where the connected sum of Σ1 and Σ2 is taken along disks D1 and D2 . Hence galt (L1 #L2 ) ≤ galt (L1 ) + galt (L2 ), and therefore the alternating genus of a link is an alternating distance. 2 In the previous proof, we showed that alternating genus is sub-additive with respect to connected sums. Balm [6] studies the behavior of alternating genus, Turaev genus, and related invariants under connect sum, and gives conditions where additivity of these invariants under connected sums is guaranteed. 2.6. Warping span Shimizu [37,38] defines the warping degree of a knot or link diagram and uses warping degree to define the warping polynomial of a knot diagram [39]. She defines the span of a knot K, denoted by spn(K), to be the minimum span of the warping polynomial for any diagram of K. We define a related invariant, called the warping span of K, that is essentially a renormalization of the span of the warping polynomial. The warping span is a half-integer valued invariant. Let D be a knot diagram with c > 0 crossings, and again let Γ be the 4-valent graph obtained from D by forgetting the “over-under” information at each crossing. An edge of D is just an edge of Γ . Choose an orientation of D, and label the edges of D by e1 , e2 , . . . , e2c where edge e1 is chosen arbitrarily and edge ei+1 follows edge ei with respect to the orientation of D. Assign a weight di to each edge ei as follows. Set d1 = 0, and set di+1 = di ± 1 according to the conventions of Fig. 3. Define the warping span of D by warp(D) = 12 max{di − dj − 1 | 1 ≤ i, j ≤ 2c}. The warping span of D does not depend on the choice of orientation or the choice of initial edge e1. In fact, any choice of weight for d1 does not change warp(D). If D does not have any crossings, i.e. D is the crossingless diagram of the unknot, then define warp(D) = 0. The warping span of a knot K, denoted by warp(K), is defined to be the minimum of warp(D) taken over all diagrams D of K. For any nontrivial knot, the warping span and the span of K are related via
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Fig. 3. Weights on either side of a crossing.
Fig. 4. The (3, 4)-torus knot together with its edge weights.
warp(K) =
1 spn(K) − 1 , 2
and for the unknot U , we have warp(U ) = 0 while spn(U ) = 0. An example of a diagram decorated with its weights is given in Fig. 4. Warping span can be extended to apply to links. Suppose that D is a diagram of a link with components C1 , . . . , C such that each component has at least one crossing. Again let Γ be the graph obtained from D by forgetting the “over-under” information at each crossing. Arbitrarily choose an orientation of D and one edge to label ek1 for each component Ck . Label the remaining edges so that eki+1 follows eki with respect to the orientation of the kth component Ck of D. Suppose that component Ck has ck edges for each k with 1 ≤ k ≤ . Assign the weight dk1 = 0 to each edge ek1 , and set dki+1 = dki ± 1 according to the conventions of Fig. 3. For each component Ck , define wk = 12 max{dki − dkj − 1 | 1 ≤ i, j ≤ ck }. Set warp(D) = max{wk | 1 ≤ k ≤ }. If D is the standard diagram of an -component unlink, then define warp(D) = 0. For any link diagram D, let D U be the disjoint union of D and the standard diagram of the unknot, and define warp(D ∪ U ) = warp(D). The warping span of the link L, denoted by warp(L), is the minimum of warp(D) where D is any diagram of L. Proposition 2.6. The warping span of a link is an alternating distance. Proof. Let D be a diagram of L. If D does not contain any crossings, then warp(D) = 0. If D contains at least one crossing, then its edges contain at least two distinct weights, and hence warp(D) ≥ 0. Moreover, if warp(D) = 0, then every component of D with a crossing contains exactly two distinct weights, which can happen if and only if D is an alternating diagram. Hence warp(L) ≥ 0 and warp(L) = 0 if and only if L is alternating. Shimizu [39] proves that warp(K1 #K2 ) ≤ max{warp(K1 ), warp(K2 )} for knots K1 and K2 . Her argument also applies to the warping span of links, and so warp(L1 #L2 ) ≤ max warp(L1 ), warp(L2 ) ≤ warp(L1 ) + warp(L2 ) for any connected sum L1 #L2 of links L1 and L2 . Hence the warping span of a link is an alternating distance. 2 Shimizu proves that changing a crossing in a knot diagram D can alter warp(D) by at most one. The proof when D is instead a link diagram is identical. Consequently,
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warp(L) ≤ dalt(L)
(2.6)
for any link L. 3. Obstructions and lower bounds The invariants under consideration are defined as a minimum over all diagrams or as a minimum over all projections to some surface. Typically, invariants of this form are difficult to compute, and so it will be useful to have obstructions and computable lower bounds for as many of the invariants as possible. The first obstruction comes from the hyperbolic geometry of the link complement. Menasco [27] proves that a prime, non-split alternating link is either a torus link or a hyperbolic link. Adams et al. [3] and Adams [4] extend this result to almost-alternating and toroidally alternating knots. Proposition 3.1. (Adams et al. [3]) Let K be a prime knot. If dalt(K) = 1 or galt (K) = 1, then K is either a torus knot or a hyperbolic knot. Since a knot has alternating genus and Turaev genus zero if and only if it is alternating, Inequality (2.5) implies the following corollary. Corollary 3.2. Let K be a prime knot. If gT (K) = 1, then K is either a torus knot or a hyperbolic knot. Our computable lower bounds arise from either Khovanov homology [22] or knot Floer homology [33,34]. The Khovanov homology of a link L, denoted by Kh(L), is a bigraded Z-module with homological grading i and polynomial (or quantum) grading j. The diagonal grading δ is defined by δ = j − 2i, and when Kh(L) is decomposed over summands with respect to the δ-grading, we write Kh(L) = δ Kh δ (L). The width of Kh(L), denoted by w(Kh(L)), is defined as 1 w Kh(L) = max δ Kh δ (L) = 0 − min δ Kh δ (L) = 0 + 1. 2 The factor of 1/2 is included in the definition of width since if L has an odd number of components, all δ-gradings where Kh δ (L) = 0 are even, and if L has an even number of components, all δ-gradings where Kh δ (L) = 0 are odd. Champanerkar and Kofman [7] and independently Wehrli [45] show that there is a complex whose generators correspond to spanning trees of the checkerboard graph of a diagram D of L and whose homology is the Khovanov homology Kh(L) of L. Champanerkar, Kofman, and Stotlzfus [9] use a relationship between spanning trees of the checkerboard graph of D and certain graphs embedded in the Turaev surface F (D) to show w Kh(L) − 2 ≤ gT (L).
(3.1)
The relationship between the Turaev surface and Khovanov homology is further explained by Dasbach and Lowrance [13]. Let F denote the vector space with two elements. We consider the “hat version” HF K(K) of the knot Floer homology of a knot K with coefficients in F. The invariant HF K(K) is a bigraded F-vector space with Maslov (or homological) grading m and Alexander (or polynomial) grading s. The diagonal grading δ is defined as δ = s − m, and when HF K(K) is decomposed over summands with respect to the δ-grading, K(K), denoted by w(HF we write HF K(K) = δ HF K δ (K). The width of HF K(K)), is defined as w HF K(K) = max δ HF K δ (K) = 0 − min δ HF K δ (K) = 0 + 1.
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Fig. 5. Left: a positive crossing. Right: a negative crossing.
Ozsváth and Szabó [32] show that there is a complex whose generators correspond to spanning trees of K(K). The author [25] uses the the checkerboard graph of a diagram D of K and whose homology is HF Ozsváth–Szabó spanning tree complex to prove that w HF K(K) − 1 ≤ gT (K).
(3.2)
Rasmussen [35] uses Lee’s spectral sequence [24] to show that the Khovanov complex of a knot K gives rise to a concordance invariant s(K). Suppose that K+ and K− are two knots such that K+ can be transformed into K− by changing a single positive crossing to a negative crossing in some diagram (see Fig. 5). Then s(K− ) ≤ s(K+ ) ≤ s(K− ) + 2.
(3.3)
Let σ(K) denote the signature of a knot with sign convention chosen so that the signature of the positive trefoil is −2. Cochran and Lickorish [10] show that σ(K− ) − 2 ≤ σ(K+ ) ≤ σ(K− ).
(3.4)
Abe [1] uses the behavior of the s-invariant and signature under crossing changes to show s(K) + σ(K) ≤ 2 alt(K).
(3.5)
Using work on knot signature of Murasugi [30] and Thistlethwaite [42] and the spanning tree complexes of Champanerkar, Kofman and Wehrli, Dasbach and the author [12] show that s(K) + σ(K) ≤ 2gT (L).
(3.6)
4. Some families of links and their alternating distances In this section we examine the three families of links: iterated Whitehead doubles of the figure-eight knot, modified torus links, and the (3, q)-torus links. We use these three families to prove Theorems 1.1 and 1.2. 4.1. Iterated Whitehead doubles of the figure-eight knot Let P be a knot embedded in a genus one handlebody Y . For any knot K, identify a regular neighborhood of K with Y such that the generator of H1 (Y, Z) is identified with a longitude of K coming from a Seifert surface. The image of P is a knot S, called a satellite of K. The knot P is the pattern for S, and K is the companion. Define the positive t-twisted Whitehead double of a knot K, denoted by D+ (K, t), to be the satellite of K where the pattern knot P is the t-twisted positive clasp knot given in Fig. 6. We use iterated Whitehead doubles of the figure-eight knot to show that the Turaev genus and dealternating number dominate the alternation number of a link.
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Fig. 6. The satellite of the figure-eight K with pattern P and companion K is the positive t-twisted Whitehead double D+ (K, t) of K. Each box labeled t indicates t positive full twists.
Let W0 be the figure-eight knot 41 . For each positive integer n, define Wn = D+ (Wn−1 , 0), that is Wn is the positive n-th iterated untwisted Whitehead double of the figure-eight knot. Define the family F(Wn) by F(Wn ) = {Wn | n ≥ 0, n ∈ Z}. Hedden [15] computes the knot Floer homology of Wn . Proposition 4.1. (Hedden [15]) Let F denote the field with two elements, and let Fk(m) denote the vector space Fk in homological grading m. Then n ⎧ 2n m n ⎪ F ⎪ m=0 (1−m) ⎪ ⎪ n ⎪ ⎪ n 2n+1 m ⎨ F F (0) m=0 (−m) HF K ∗ (Wn , s) ∼ = n ⎪ ⎪ 2n m n ⎪ ⎪ F(−1−m) ⎪ m=0 ⎪ ⎩ 0
s = 1, s = 0, s = −1, otherwise,
and w(HF K(Wn )) = n + 1. The alternation number, Turaev genus, and alternating genus of Wn behave according to the following proposition. Proposition 4.2. For each positive integer n, alt(Wn ) = 1, gT (Wn ) ≥ n,
and
galt (Wn ) > 1. Proof. Let n be a positive integer. Changing one of the crossings of the clasp in any Whitehead double transforms the knot into an unknot, and thus the unknotting number u(Wn) is one. Since the unknot is alternating, the inequality alt(K) ≤ u(K) holds for every knot, and hence alt(Wn ) ≤ 1. The knot Wn is non-alternating since w(HF K(Wn )) > 1, and so alt(Wn ) = 1. Inequality (3.2) states that for any knot K, we have w(HF K(K)) −1 ≤ gT (K), and hence Proposition 4.1 implies that gT (Wn ) ≥ n.
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Since the genus of Wn is one, it follows that Wn is prime, and because Wn is a satellite knot, Proposition 3.1 implies that galt (Wn ) = 1. Since Wn is non-alternating, we may conclude that galt (Wn ) > 1. 2 4.2. Modified torus links The modified torus link T(p, q) is obtained by changing certain crossings of a standard diagram of the (p, q)-torus link T (p, q). We use the natural embedding of T (p, q) on a torus to show that T(p, q) is toroidally alternating for many choices of p and q. The behavior of the Rasmussen s-invariant and knot signature under crossing changes imply that the Turaev genus and alternation number of T(p, q) can be arbitrarily large. p denote Let Bp denote the p-stranded braid group, let Δp ∈ Bp denote the braid σ1 σ2 · · · σp−1 , and let Δ the braid p = Δ
p−1
(−1)i+1
σi
(−1)p
= σ1 σ2−1 σ3 · · · σp−1 .
i=1
Define T(p, q) to be the closure of the braid Δq−1 p Δp . The link T (p, q) can be obtained from the diagram p−1 q of the closure of Δp by changing 2 crossings where the closure of Δqp is a familiar diagram of the (p, q)-torus link T (p, q). Define the family F(T(p, q)) by F T(p, q) = T(p, q) p, q ≥ 3, p, q ∈ Z . Gordon, Litherland, and Murasugi [14] give the following recursive algorithm for computing the signature of the torus link T (p, q). Theorem 4.3. (Gordon et al. [14]) Suppose that p, q > 0. The following recurrence formulas hold. (1) Suppose that 2p < q. (a) If p is odd, then σ(T (p, q)) = σ(T (p, q − 2p)) − p2 + 1. (b) If p is even, then σ(T (p, q)) = σ(T (p, q − 2p)) − p2 . (2) σ(T (p, 2p)) = 1 − p2 . (3) Suppose that p ≤ q < 2p. (a) If p is odd, then σ(T (p, q)) = 1 − p2 − σ(T (p, 2p − q)). (b) If p is even, then σ(T (p, q)) = 2 − p2 − σ(T (p, 2p − q)). (4) σ(T (p, q)) = σ(T (q, p)), σ(T (p, 1)) = 0, σ(T (2, q)) = 1 − q. In order to estimate the bounds in Inequalities (3.5) and (3.6), we first estimate the signature and Rasmussen invariant for torus knots and the modified torus knots T(p, q). Proposition 4.4. Let p and q be relatively prime integers with p ≥ 3 and q ≥ 3. Then 1 1 −(p − 1)(p − 2) − pq ≤ σ T (p, q) ≤ (p − 1)(p − 2) − (p − 1)q, 2 2 1 1 −(p − 1)(p − 2) − pq ≤ σ T(p, q) ≤ (p − 1)2 − (p − 1)q, and 2 2 pq − 2p − q + 2 ≤ s T (p, q) ≤ pq − p − q + 1.
(4.1) (4.2) (4.3)
Proof. Let k be an integer relatively prime to p with 0 < k < p. Rudolph [36] shows that the signature of the closure of a positive braid is negative, and hence σ(T (p, k)) ≤ 0. The unknotting number u(K) of any
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knot K satisfies |σ(K)| ≤ 2u(K). Kronheimer and Mrowka [23] show that the unknotting number of T (p, k) is 12 (p − 1)(k − 1), and thus −(p − 1)(p − 2) ≤ σ T (p, k) ≤ 0.
(4.4)
Let q = np + r where n is a non-negative integer and 0 < r < p. Suppose that n = 0. Then (p − 1)(p − 2) − 12 (p − 1)q = (p − 1)(p − 2) − 12 (p − 1)r ≥ 0 because 12 r ≤ p − 2. Thus 1 1 −(p − 1)(p − 2) − pq = −(p − 1)(p − 2) − pr 2 2 < −(p − 1)(p − 2) ≤ σ T (p, q) ≤0 1 ≤ (p − 1)(p − 2) − (p − 1)q, 2 and Inequality (4.1) is proven when n = 0. Now suppose that n > 0. Repeated applications of Theorem 4.3 yields
σ T (p, np + r) =
⎧ σ(T (p, r)) − 12 n(p2 − 1) ⎪ ⎪ ⎪ ⎪ ⎨ σ(T (p, r)) − 1 np2 2 ⎪ −σ(T (p, p − r)) − ⎪ ⎪ ⎪ ⎩ −σ(T (p, p − r)) −
1 2 (n 1 2 (n
p odd, n even, p even, n even,
+ 1)(p2 − 1) p odd, n odd, + 1)p2 + 2
(4.5)
p even, n odd.
Since 0 < r < p and 0 < p − r < p, Inequality (4.4) implies that −(p − 1)(p − 2) ≤ σ(T (p, r)) ≤ 0 and −(p − 1)(p − 2) ≤ σ(T (p, p − r)) ≤ 0. Combining these inequalities with Eq. (4.5) yields the inequalities 1 1 −(p − 1)(p − 2) − np2 − 1 ≤ σ T (p, q) ≤ (p − 1)(q − 1) − (n + 1) p2 − 1 . 2 2 Since q = np + r, it follows that pq = np2 + rp > np2 + 2, and hence 1 1 −(p − 1)(p − 2) − pq < −(p − 1)(p − 2) − np2 − 1 ≤ σ T (p, q) . 2 2 Likewise, since q = np + r, it follows that (p − 1)q = np2 + rp − np − r ≤ np2 + p2 − n − 1 = (n + 1) p2 − 1 . Hence 1 1 σ T (p, q) ≤ (p − 1)(q − 1) − (n + 1) p2 − 1 ≤ (p − 1)(p − 2) − (p − 1)q, 2 2 and thus Inequality (4.1) is proven for all n. Since T(p, q) can be obtained from T (p, q) via p−1 2 crossing changes, Inequality (4.2) follows from Inequalities (3.4) and (4.1). Rasmussen computes the s-invariant for positive knots, and applying his formula to T (p, q) yields s(T (p, q)) = pq − p − q + 1. Inequality (4.3) follows from this fact and Inequality (3.3). 2
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Corollary 4.5. Let p be a fixed integer with p ≥ 3. For any positive integer n, there exists a q relatively prime to p such that alt(T(p, q)) ≥ n and gT (T(p, q)) ≥ n. Proof. Proposition 4.4 implies that if p and q are relatively prime with p ≥ 3 and q ≥ 3 then 1 pq − 2p − q + 2 − (p − 1)(p − 2) ≤ s T(p, q) + σ T(p, q) 2 If p is fixed and q goes to infinity, then the left hand side of the previous inequality is eventually positive and grows without bound. Therefore for sufficiently large values of q (with p and q relatively prime), we have 1 pq − 2p − q + 2 − (p − 1)(p − 2) ≤ s T(p, q) + σ T(p, q) . 2 Hence Inequalities (3.5) and (3.6) imply that alt(T(p, q)) and gT (T(p, q)) can be arbitrarily large. 2 An alternate proof that gT (T(p, q)) can be arbitrarily large uses Inequality (3.1) and a computation of the Khovanov width of T(p, q). While the alternation number and Turaev genus of T(p, q) grow without bound, the next proposition shows that many such knots are either alternating or toroidally alternating. Proposition 4.6. Let p be an even integer and let q be an odd integer with p ≥ 4, q ≥ 3, and q = 1 mod p. Then galt T(p, q) ≤ 1. Proof. Let Σ be a Heegaard torus in S 3 and suppose that T (p, q − 1) is embedded on Σ by going p times around the longitude and q − 1 times around the meridian. Cut Σ along a meridian to obtain a cylinder C1 . p is embedded on a cylinder C2 so that the incoming strands meet Suppose a standard planar diagram of Δ one boundary component and the outgoing strands meet the other. Identify each boundary component of C1 with a boundary component of C2 so that the resulting surface Σ is again a Heegaard torus in S 3 with a diagram of T(p, q) projected onto it (via some projection π). Before we glue in the cylinder C2 each component of Σ − T (p, q − 1) is an annulus, and since q = p mod 1, p , the the number of components of Σ − T (p, q − 1) is strictly less than p. By surgering in the diagram of Δ annuli are separated into a disjoint union of disks. p by 1, 2, . . . , p from left to right. The incoming strands Label the incoming and outgoing strands of Δ labeled with odd numbers encounter an under-crossing first, while the incoming strands labeled with even integers encounter an over-crossing first. For the outgoing strands, the situation is reversed. Outgoing strands labeled with odd numbers most recently encountered over-crossings, while outgoing strands labeled with even integers most recently encountered under-crossings. Since p is even and q − 1 is even, the permutation on the strands induced by Δq−1 sends strands labeled with odd numbers to strands labeled with odd p numbers and sends strands labeled with even numbers to strands labeled with even numbers. Therefore, the projection π(T(p, q)) is alternating on Σ. Since T(p, q) has an alternating projection to a Heegaard torus where the complement of the projection is a disjoint union of disks, it follows that galt (T(p, q)) ≤ 1. 2 Fig. 7 shows that T(4, 3) and T(4, 4k + 3) for non-negative k have alternating projections to Heegaard tori in S 3 whose complements are disjoint unions of disks. The examples in Fig. 7 demonstrate that there is a Heegaard surface Σ on which the knot has an alternating projection whose complement is a collection of disks such that Σ is not the Turaev surface of any diagram of the knot. Armond, Druivenga, and Kindred [5] show how to ensure such a surface is a Turaev surface using Heegaard diagrams.
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Fig. 7. After identifying the two components of the boundary of each cylinder to obtain tori, the diagram on the left is an alternating projection of T (4, 3) to the torus, and the diagram on the right is an alternating projection of T (4, 4k + 3) to the torus for each k ≥ 0.
4.3. The (3, q)-torus knots Let F(T (3, q)) be the family of (3, q)-torus links where q is any integer. The (3, q)-torus knots have arbitrarily large alternation number, Turaev genus, c(T (3, q)) − spanT (3,q) (t), and dealternating number, but have warping span 0 or 1/2. Kanenobu [18] computes the alternation numbers of the (3, q)-torus knots, up to an additive error of at most one. Proposition 4.7. (Kanenobu [18]) For any positive integer n, alt T (3, 4) = alt T (3, 5) = 1, alt T (3, 6n + 1) = alt T (3, 6n + 2) = 2n, alt T (3, 6n + 4) = alt T (3, 6n + 5) = 2n or 2n + 1. Using Inequality (3.1) and work of Stošić [40] and Turner [44], the author [26] computes the Turaev genus of the (3, q)-torus knots. Abe and Kishimoto [2] independently compute the Turaev genus of the (3, q)-torus knots and also compute their dealternating numbers. Proposition 4.8. (Abe and Kishimoto [2], Lowrance [26]) Let n be a non-negative integer, and let i = 1 or 2. Then gT T (3, 3n + i) = dalt T (3, 3n + i) = n. Combining work of Jones and Murasugi, we obtain the following result about the difference between the crossing number and the span of the Jones polynomial of T (3, q). Proposition 4.9. (Jones [17], Murasugi [31]) Suppose that q is relatively prime to 3 and q > 3. Then c T (3, q) − span VT (3,q) (t) = q − 1.
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Fig. 8. The 3-braids σ1 σ2 (left) and σ1−1 σ2−1 (right) with their edges labeled by weights.
Proof. Jones [17] gives the following formula for the Jones polynomial of all torus knots:
VT (p,q) (t) =
t(p−1)(q−1)/2 1 − tp+1 − tq+1 + tp+q . 2 1−t
For the (3, q)-torus knots, this formula yields VT (3,q) (t) = tq−1 + tq+1 − t2q . Hence span VT (3,q) (t) = q + 1. Murasugi [31] shows that the crossing number of T (3, q) where q > 3 is 2q. Therefore c T (3, q) − span VT (3,q) (t) = q − 1.
2
An example of Shimizu [39] is easily generalized to the following result. Proposition 4.10. If q is an integer with |q| > 2, then warp(T (3, q)) = 12 . Proof. Suppose q > 0. We write T (3, q) as the closure of (σ1 σ2 )q , and T (3, −q) as the closure of (σ1−1 σ2−1 )q . Fig. 8 depicts the 3-braids σ1 σ2 and σ1−1 σ2−1 with edges labeled by weights 0, 1, and 2. Since the incoming weights are the same as the outgoing weights, these braids can be stacked q times to obtain diagrams of T (3, q) and T (3, −q) where the only weights are 0, 1, and 2. Hence warp(T (3, q)) = warp(T (3, −q)) ≤ 12 . Since T (3, q) is alternating if and only if |q| ≤ 2, the result follows. 2
4.4. Proofs of Theorems 1.1 and 1.2 The proofs of Theorems 1.1 and 1.2 are pieced together from the previous three subsections. Proof of Theorem 1.1. Proposition 4.2 shows that gT (Wn ) −alt(Wn ) ≥ n −1 and Inequality (2.4) shows that dalt(Wn ) − alt(Wn ) ≥ n − 1. Thus alt(F(Wn )) gT (F(Wn )) and alt(F(Wn )) dalt(F(Wn )). Inequality (2.3) implies that c(L) − span VL (t) dominates the alternation number on F(Wn ). Fix an even integer p ≥ 4 and let q range over all positive integers such that p and q are relatively prime, and q = 1 mod p. Proposition 4.6 states that galt (T(p, q)) = 1 while Corollary 4.5 implies that alt(T(p, q)) and gT (T(p, q)) go to infinity as q goes to infinity. Hence galt (F(T(p, q))) alt(F(T(p, q))) and galt (F(T(p, q))) gT (F(T(p, q))). Both Inequality (2.1) and Inequality (2.4) imply that galt (F(T(p, q))) dalt(F(T(p, q))). Inequality (2.3) implies that the difference c(L) − span VL (t) dominates the alternating genus on F(T(p, q)).
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Fig. 9. The crossing to the left is extended by the alternating tangle to the right.
Now let q be a positive integer not divisible by 3 with q > 3. Propositions 4.7, 4.8, 4.9, and 4.10 yield c T (3, q) − span VT (3,q) (t) = q − 1, dalt T (3, q) = q/3, gT T (3, q) = q/3, alt T (3, q) = q/3 or q/3 − 1, warp T (3, q) = 1/2,
and
which implies statements (3) and (4). 2 Proof of Theorem 1.2. Proposition 4.2 implies that for the positive n-th iterated untwisted Whitehead double Wn of the figure-eight knot, alt(Wn ) < galt (Wn ) for each positive integer n. Furthermore, we saw in the proof of Theorem 1.1 that galt (F(T(p, q))) alt(F(T(p, q))). Therefore the alternation number and alternating genus of a link are not comparable. 2 5. Further questions Abe and Kishimoto [2] showed that gT (D) ≤ dalt(D) for any link diagram D. For many diagrams, this inequality is strict, however the following question remains open. Question 1. Is there a link L such that gT (L) < dalt(L)? We now describe a method to construct diagrams where dalt(D) − gT (D) is arbitrarily large. Let x be a crossing in a link diagram D considered as a 2-tangle. An alternating 2-tangle τ is said to extend the crossing x, if there is some choice of resolutions of all but one crossing of τ such that the resulting tangle is isotopic to x through a planar isotopy fixing the endpoints of the tangle. If D is a diagram with crossing x that is extended by the rational tangle τ , then let D(x, τ ) be the diagram obtained by replacing the 2-tangle x with τ (see Fig. 9). Let D be a link diagram with c(D) crossings such that dalt(D) = k. Suppose that x is one of the k crossings changed to make D alternating, and suppose that τ is an alternating tangle with c(τ ) crossings extending x. In order to minimally transform D(x, τ ) into an alternating diagram, one must either change the k − 1 other crossing from above along with every crossing of τ or one must change every crossing other than the k crossings from above. Hence dalt D(x, τ ) = min c(D) − k, c(τ ) + k − 1 . One can show that |sA (D)| + |sB (D)| − c(D) = |sA (D(x, τ ))| + |sB (D(x, τ ))| − c(D(x, τ )), and therefore gT (D(x, τ )) = gT (D). Hence a suitably chosen sequence of alternating tangle extensions can force the gap
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between the dealternating number of a diagram and genus of the Turaev surface of the diagram to grow without bound. Question 1 being open means that it is not yet determined whether the dealternating number and the Turaev genus are actually different invariants. It is natural to ask whether Inequality (2.3) has an analog for the dealternating number. Question 2. Is dalt(L) ≤ c(L) − span VL (t) for any link L. Turaev [43] shows that gT (D) ≤ c(D) − span VL (t) for any diagram D of a link L. However, this approach does not work for the dealternating number because there exists a diagram where dalt(D) > c(D) − span VL (t). Let D be the usual diagram (5, −3, 2) pretzel knot, and let P (5, −3, 2) be the knot with diagram D. Then P (5, −3, 2) is knot 10125 in the Rolfsen table. We have that dalt(D) = 3 while c(D) = 10 and span VP (5,−3,2) (t) = 8. Thus dalt(D) = 3 > 2 = c(D) − span VP (5,−3,2) (t). Kim and Lee [21] prove that every non-alternating pretzel link has dealternating number one, and hence P (5, −3, 2) has a some diagram (not D) implying that dalt(P (5, −3, 2)) = 1. By examining the behavior of Whitehead doubles, we saw that the Turaev genus dominates the alternation number of a link. However, the following question remains open. Question 3. Are the alternation number and the Turaev genus of a link comparable? If the alternation number and Turaev genus of a link are comparable, then we must have alt(L) ≤ gT (L) for any link L. If they are not comparable, then there exists a link L such that gT (L) < alt(L), and hence Inequality (2.1) would imply an affirmative answer to Question 1. Inequality (2.6) follows from the fact that warp(D) ≤ dalt(D) for every link diagram D. Since Turaev genus dominates the warping span, the following question is natural. Question 4. Are the warping span and the Turaev genus of a link comparable? If warping span and Turaev genus are comparable, then warp(L) ≤ gT (L) for any link L. However, this cannot be proved in the same way as Inequality (2.6) because there exist diagrams D with warp(D) > gT (D). For example, let D be the diagram of the unknot obtained by taking the closure of the 2-braid σ1−1 σ1 σ1−1 σ1 σ1−1 . A straightforward computation shows that warp(D) = 2 while gT (D) = 1. If the warping span and Turaev genus are not comparable, then there exists a link L with gT (L) < warp(L). In order to confirm such an example, it would be useful to find lower bounds for the warping span of a link. The case of alternating genus is similar. Proposition 3.1 gives an obstruction for certain links to have alternating genus one, but beyond that there is no useful lower bound for the alternating genus of a link. This leads naturally to our final question. Question 5. Do links with arbitrarily large alternating genus or warping span exist? Acknowledgements The author is grateful for comments from Oliver Dasbach, Radmila Sazdanović, and Alexander Zupan. References [1] Tetsuya Abe, An estimation of the alternation number of a torus knot, J. Knot Theory Ramif. 18 (3) (2009) 363–379. [2] Tetsuya Abe, Kengo Kishimoto, The dealternating number and the alternation number of a closed 3-braid, J. Knot Theory Ramif. 19 (9) (2010) 1157–1181.
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Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Alternating distances of knots and links Adam M. Lowrance 1 Department of Mathematics, Vassar College, Poughkeepsie, NY, United States
a r t i c l e
i n f o
Article history: Received 26 June 2014 Accepted 7 December 2014 Available online xxxx Keywords: Alternating knot Dealternating Almost alternating Alternation Turaev genus Alternating genus Toroidally alternating Jones polynomial Warping polynomial Warping degree Knot theory Khovanov homology Knot Floer homology Homological width
a b s t r a c t An alternating distance is a link invariant that measures how far away a link is from alternating. We study several alternating distances and demonstrate that there exist families of links for which the difference between certain alternating distances is arbitrarily large. We also show that two alternating distances, the alternation number and the alternating genus, are not comparable. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Alternating links play an important role in knot theory and 3-manifold geometry and topology. Link invariants are often easier to compute and take on special forms for alternating links. Moreover, the complements of alternating links have interesting topological and geometric structures. Many generalizations of alternating links exist, and a particular generalization can give rise to an invariant that measures how far a link is from alternating. We study several such invariants, which we call alternating distances. A link L is split if it has a separating sphere, i.e. a two-sphere S 2 in S 3 such that L and S 2 are disjoint and each component of S 3 − S 2 contains at least one component of L. We will mostly be concerned with non-split links, that is links with no separating spheres. A real valued link invariant d(L) is an alternating distance if it satisfies the following conditions.
1
E-mail address: [email protected]. The author was supported by an AMS–Simons Travel Grant.
http://dx.doi.org/10.1016/j.topol.2014.12.010 0166-8641/© 2014 Elsevier B.V. All rights reserved.
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(1) For any non-split link, d(L) ≥ 0. (2) For any non-split link, d(L) = 0 if and only if L is alternating. (3) If L1 and L2 are non-split links and L1 #L2 is any connected sum of L1 and L2 , then d(L1 #L2 ) ≤ d(L1 ) + d(L2 ). The connected sum L1 #L2 depends on a choice of components in L1 and L2 . However, condition (3) above must be true for any choice of connected sum. We will frequently use the notation L1#L2 to denote an arbitrary choice of connected sum. We consider the following invariants. The dealternating number, denoted by dalt(L), and the alternation number, denoted by alt(L), are defined by counting crossing changes. The relationship between the minimum crossing number c(L) of a link L and the span of the Jones polynomial VL (t) of L was used to prove some of Tait’s famous conjectures, and we study the difference c(L) − span VL (t). The Turaev genus, denoted by gT (L), and the alternating genus, denoted by galt (L), are the genera of certain surfaces associated to L. The warping span, denoted by warp(K), is defined by examining the over-under behavior as one travels along the knot or link. Precise definitions of these invariants are given in Section 2. Let d1 and d2 be real valued link invariants and let F be a family of links. We say d2 dominates d1 on F, and write d1 (F) d2 (F), if for each positive integer n, there exists a link Ln ∈ F such that d2 (Ln ) − d1 (Ln ) ≥ n. In Section 4, we examine three families of links. The first family F(Wn ) consists of iterated Whitehead doubles of the figure-eight knot. The second family F(T(p, q)) consists of links obtained by changing certain crossings of torus links. The third family F(T (3, q)) consists of the (3, q)-torus knots. Theorem 1.1. Let F(Wn ), F(T(p, q)), and F(T (3, q)) be the families of links above. (1) The dealternating number, c(L) − span VL (t), and Turaev genus dominate the alternation number on F(Wn ). (2) The dealternating number, Turaev genus, c(L) − span VL (t), and alternation number dominate the alternating genus on F(T(p, q)). (3) The dealternating number, alternation number, c(L) −span VL (t), and Turaev genus dominate the warping span on F(T (3, q)). (4) The difference c(L) − span VL (t) dominates the dealternating number, alternation number, alternating genus, and Turaev genus on F(T (3, q)). Two real valued link invariants d1 and d2 are said to be comparable if either d1 (L) ≤ d2 (L) or d2 (L) ≤ d1 (L) for all links L. The invariants d1 and d2 are not comparable if there exist links L and L such that d1 (L) < d2 (L) and d1 (L ) > d2 (L ). Theorem 1.2. The alternation number and alternating genus of a link are not comparable. This paper is organized as follows. In Section 2, we define the invariants mentioned in Theorems 1.1 and 1.2. In Section 3, we describe some lower bounds for the invariants. In Section 4, we define several families of links and use them to prove Theorems 1.1 and 1.2. In Section 5, we discuss some open questions about our invariants. 2. The invariants In this section, the invariants of Theorems 1.1 and 1.2 are defined. We show that each one is an alternating distance, and discuss some known relationships between them.
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2.1. Dealternating number Adams et al. [3] define almost alternating links to be non-alternating links with a diagram such that one crossing change makes the diagram alternating. They use the notion of number of crossing changes needed to make a diagram alternating to define the dealternating number of a link. The dealternating number of a link diagram D, denoted by dalt(D), is the minimum number of crossing changes necessary to transform D into an alternating diagram. The dealternating number of a link L, denoted by dalt(L), is the minimum dealternating number of any diagram of L. A link with dealternating number k is also called k-almost alternating. Proposition 2.1. The dealternating number of a link is an alternating distance. Proof. The definition of the dealternating number implies that it is always non-negative and equals zero if and only if the link is alternating. Suppose that L1 and L2 are links with diagrams D1 and D2 such that dalt(D1 ) = dalt(L1 ) and dalt(D2 ) = dalt(L2 ). For any choice of connected sum L1 #L2 , there exists some choice D1 #D2 of connected sum of diagrams D1 and D2 such that D1 #D2 is a diagram of L1 #L2 and dalt(D1 #D2 ) = dalt(D1 ) + dalt(D2 ) = dalt(L1 ) + dalt(L2 ). Thus dalt(L1 #L2 ) ≤ dalt(L1 ) + dalt(L2 ) and hence the dealternating number of a link is an alternating distance. 2 2.2. Alternation number Kawauchi [20] uses crossing changes in a slightly different manner to define the alternation number of a link. The alternation number of a link diagram D, denoted by alt(D), is the minimum number of crossing changes necessary to transform D into some (possibly non-alternating) diagram of an alternating link. The alternation number of a link L, denoted by alt(L), is the minimum alternation number of any diagram of L. The alternation number of L is also the Gordian distance from L to the set of alternating links [28]. It is immediate from their definitions that alt(L) ≤ dalt(L)
(2.1)
for any link L. Proposition 2.2. The alternation number of a link is an alternating distance. Proof. The definition of the alternation number implies that it is always non-negative and equals zero if and only if the link is alternating. Suppose that L1 and L2 are links with diagrams D1 and D2 respectively such 1 and D 2 be the diagrams of alternating links obtained that alt(D1 ) = alt(L1 ) and alt(D2 ) = alt(L2 ). Let D from D1 and D2 respectively via the minimum number of crossing changes. Let L1 #L2 be a connected sum of L1 and L2 , and let D1 #D2 be a connected sum of D1 and D2 such that D1 #D2 is a diagram of L1 #L2 . 2 (an alternating link) via alt(D1 ) +alt(D2 ) = alt(L1 ) +alt(L2 ) 1 #D Then D1 #D2 can be transformed into D crossing changes. Hence alt(L1 #L2 ) ≤ alt(L1 ) + alt(L2 ), and thus the alternation number is an alternating distance. 2 2.3. Crossing number and the span of the Jones polynomial In the late 19th century, Tait [41] conjectured that a certain type of alternating link diagram (called reduced) has minimal crossing number among all diagrams for that link. This conjecture remained undecided until the discovery of the Jones polynomial [16], when combined work of Kauffman [19] and Murasugi [29] proved the conjecture to be true.
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Fig. 1. The A and B resolutions of a crossing.
Fig. 2. In a neighborhood of each vertex of Γ a saddle surface transitions between the all-A and all-B states.
Let VL (t) denote the Jones polynomial of L, and let max deg VL (t) and min deg VL (t) denote the maximum and minimum power of t in VL (t) with non-zero coefficient respectively. Define span VL (t) = max deg VL (t) − min deg VL (t). Let c(L) denote the minimum number of crossings in any diagram of the link L. Proposition 2.3. The difference c(L) − span VL (t) is an alternating distance. Proof. Murasugi [29] and Kaufman [19] prove that span VL (t) ≤ c(L) for any non-split link L. Furthermore, when L is non-split, Murasugi proves that span VL (t) = c(L) if and only if L is a connected sum of alternating links, which happens if and only if L is alternating. The span of the Jones polynomial is additive under connected sums, i.e. span VL1 #L2 (t) = span VL1 (t) +span VL2 (t). It is a long-standing open question whether crossing number is additive, but it is easy to see that crossing number is sub-additive, i.e. c(L1 #L2 ) ≤ c(L1 ) + c(L2 ). Therefore c(L1 #L2 ) − span VL1 #L2 (t) ≤ (c(L1 ) − span VL1 (t)) + (c(L2 ) − span VL2 (t)). Hence c(L) − span VL (t) is an alternating distance. 2 2.4. Turaev genus Turaev [43] gives a simplified proof of Tait’s conjecture mentioned above, where he associates to each link diagram D a surface F (D), now known as the Turaev surface of D. Each crossing of D can be resolved in either an A-resolution or a B-resolution, as depicted in Fig. 1. A collection s of simple closed curves obtained by choosing either an A-resolution or a B-resolution for each crossing of D is a state of D. The state sA (D) obtained by choosing an A-resolution at each crossing is called the all-A state of D, and similarly, the state sB (D) obtained by choosing a B-resolution at each crossing is called the all-B state of D. Let Γ be the 4-valent graph obtained from D by forgetting the “over-under” information at each crossing. Regard Γ as embedded in S 2 , and thicken the sphere to S 2 × [−1, 1] such that Γ is a subset of S 2 × {0}. We first describe the intersection of the Turaev surface F (D) and S 2 × [−1, 1]. Outside of neighborhoods of the vertices of Γ , the Turaev surface intersects S 2 × [−1, 1] in Γ × [−1, 1]. In a neighborhood of each crossing, the Turaev surface intersects S 2 × [−1, 1] in a saddle positioned so that F (D) ∩ S 2 × {−1} is the all-B state sB (D) and F (D) ∩ S 2 × {1} is the all-A state sA (D), as depicted in Fig. 2. Summarizing, the intersection of the Turaev surface with S 2 × [−1, 1] is a cobordism between sA (D) and sB (D) whose saddle points correspond to the crossings of D. Outside of S 2 × [−1, 1] the Turaev surface F (D) is a collection of disks capping off the components of sA (D) and sB (D).
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The Turaev surface F (D) is always oriented, and if D is a diagram of a non-split link L, then F (D) is connected. The Turaev genus gT (D) of a diagram D is defined to be the genus of the Turaev surface F (D). It can be shown that gT (D) =
1 2 + c(D) − sA (D) − sB (D) , 2
(2.2)
where c(D) is the number of crossings of D, and |sA (D)| and |sB (D)| denote the number of components in the all-A and all-B states of D. The Turaev genus gT (L) of a non-split link L is the minimum genus of the Turaev surface of D where D is any diagram of L, i.e. gT (L) = min gT (D) D is a diagram of L . A closed, oriented surface Σ ⊂ S 3 is a Heegaard surface if both components of S 3 − Σ are handlebodies. If Σ is a Heegaard surface in S 3 , then every link L has an isotopy class representative that lies within a thickened neighborhood Σ × [−1, 1] of Σ. Let π : Σ × [−1, 1] → Σ be projection onto the first factor. The link L is said to have an alternating projection to Σ if π(L) has only transverse double points and the crossings of the projection π(L) alternate as one travels along each component of L. Dasbach et al. [11] prove the following facts about the Turaev surface F (D) and Turaev genus of a non-split link. • • • •
The The The The
Turaev surface F (D) is a Heegaard surface in S 3 . link L has an alternating projection π to F (D). complement F (D) − π(L) of the projection π is a disjoint union of disks. Turaev surface of an alternating diagram is a sphere, and gT (L) = 0 if and only if L is alternating.
Proposition 2.4. The Turaev genus of a link is an alternating distance. Proof. Since Turaev genus is a minimum genus of a surface, it is always non-negative. As mentioned above, a link L has Turaev genus zero if and only if it is alternating. If D1 and D2 are link diagrams and D1 #D2 is any connected sum, then |sA (D1 #D2 )| = |sA (D1 )| + |sA (D2 )| − 1 and |sB (D1 #D2 )| = |sB (D1 )| + |sB (D2 )| − 1. Then Eq. (2.2) implies that gT (D1 #D2 ) = gT (D1 ) +gT (D2 ). Hence if L1 and L2 are links and L1 #L2 is any connected sum, then gT (L1 #L2 ) ≤ gT (L1 ) + gT (L2 ). Therefore, the Turaev genus of a link is an alternating distance. 2 Turaev [43] shows that gT (D) ≤ c(D) − span VL (t) for any diagram D of the link L. Minimizing over all diagrams of the link L, one obtains gT (L) ≤ c(L) − span VL (t).
(2.3)
Abe and Kishimoto [2] examine the behavior of the Turaev surface under crossing changes to show that gT (D) ≤ dalt(D) for any link diagram D. Consequently, for any link L, gT (L) ≤ dalt(L).
(2.4)
Champanerkar and Kofman’s recent survey [8] gives many open questions concerning the Turaev genus of a link.
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2.5. Alternating genus Following his work on almost alternating links, Adams [4] defined toroidally alternating links as those links L that have an alternating projection π to a Heegaard torus Σ such that the complement of the projection in the Heegaard torus, i.e. Σ − π(L), is a disjoint union of disks. Toroidally alternating links can be naturally generalized. Define the alternating genus of a non-split link L, denoted by galt (L), to be the minimum genus of any Heegaard surface Σ such that the link L lies within a regular neighborhood Σ × [−1, 1] of the Heegaard surface, has an alternating projection π : Σ × [−1, 1] → Σ, and Σ − π(L) is a disjoint union of disks. For any diagram D of L, the Turaev surface F (D) is an example of such a surface, and so galt (L) is well-defined, and galt (L) ≤ gT (L)
(2.5)
for any link L. Proposition 2.5. The alternating genus of a link is an alternating distance. Proof. By definition, a non-split link L has alternating genus zero if and only if it has an alternating projection to a sphere, i.e. if and only if L is alternating. Moreover, since alternating genus is the minimum genus of some surface, it is always non-negative. Let L1 #L2 be a connected sum of links L1 and L2 such that both L1 and L2 have an alternating projections π1 and π2 to respective Heegaard surfaces Σ1 and Σ2 . Moreover, suppose that both Σ1 − π1 (L1 ) and Σ2 − π2 (L2 ) are disjoint unions of disks. Then there exist disks D1 and D2 in Σ1 and Σ2 meeting the projections π(L1 ) and π(L2 ) in a single arc such that L1 #L2 has an alternating projection to Σ1 #Σ2 whose complementary regions are a disjoint union of disks where the connected sum of Σ1 and Σ2 is taken along disks D1 and D2 . Hence galt (L1 #L2 ) ≤ galt (L1 ) + galt (L2 ), and therefore the alternating genus of a link is an alternating distance. 2 In the previous proof, we showed that alternating genus is sub-additive with respect to connected sums. Balm [6] studies the behavior of alternating genus, Turaev genus, and related invariants under connect sum, and gives conditions where additivity of these invariants under connected sums is guaranteed. 2.6. Warping span Shimizu [37,38] defines the warping degree of a knot or link diagram and uses warping degree to define the warping polynomial of a knot diagram [39]. She defines the span of a knot K, denoted by spn(K), to be the minimum span of the warping polynomial for any diagram of K. We define a related invariant, called the warping span of K, that is essentially a renormalization of the span of the warping polynomial. The warping span is a half-integer valued invariant. Let D be a knot diagram with c > 0 crossings, and again let Γ be the 4-valent graph obtained from D by forgetting the “over-under” information at each crossing. An edge of D is just an edge of Γ . Choose an orientation of D, and label the edges of D by e1 , e2 , . . . , e2c where edge e1 is chosen arbitrarily and edge ei+1 follows edge ei with respect to the orientation of D. Assign a weight di to each edge ei as follows. Set d1 = 0, and set di+1 = di ± 1 according to the conventions of Fig. 3. Define the warping span of D by warp(D) = 12 max{di − dj − 1 | 1 ≤ i, j ≤ 2c}. The warping span of D does not depend on the choice of orientation or the choice of initial edge e1. In fact, any choice of weight for d1 does not change warp(D). If D does not have any crossings, i.e. D is the crossingless diagram of the unknot, then define warp(D) = 0. The warping span of a knot K, denoted by warp(K), is defined to be the minimum of warp(D) taken over all diagrams D of K. For any nontrivial knot, the warping span and the span of K are related via
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Fig. 3. Weights on either side of a crossing.
Fig. 4. The (3, 4)-torus knot together with its edge weights.
warp(K) =
1 spn(K) − 1 , 2
and for the unknot U , we have warp(U ) = 0 while spn(U ) = 0. An example of a diagram decorated with its weights is given in Fig. 4. Warping span can be extended to apply to links. Suppose that D is a diagram of a link with components C1 , . . . , C such that each component has at least one crossing. Again let Γ be the graph obtained from D by forgetting the “over-under” information at each crossing. Arbitrarily choose an orientation of D and one edge to label ek1 for each component Ck . Label the remaining edges so that eki+1 follows eki with respect to the orientation of the kth component Ck of D. Suppose that component Ck has ck edges for each k with 1 ≤ k ≤ . Assign the weight dk1 = 0 to each edge ek1 , and set dki+1 = dki ± 1 according to the conventions of Fig. 3. For each component Ck , define wk = 12 max{dki − dkj − 1 | 1 ≤ i, j ≤ ck }. Set warp(D) = max{wk | 1 ≤ k ≤ }. If D is the standard diagram of an -component unlink, then define warp(D) = 0. For any link diagram D, let D U be the disjoint union of D and the standard diagram of the unknot, and define warp(D ∪ U ) = warp(D). The warping span of the link L, denoted by warp(L), is the minimum of warp(D) where D is any diagram of L. Proposition 2.6. The warping span of a link is an alternating distance. Proof. Let D be a diagram of L. If D does not contain any crossings, then warp(D) = 0. If D contains at least one crossing, then its edges contain at least two distinct weights, and hence warp(D) ≥ 0. Moreover, if warp(D) = 0, then every component of D with a crossing contains exactly two distinct weights, which can happen if and only if D is an alternating diagram. Hence warp(L) ≥ 0 and warp(L) = 0 if and only if L is alternating. Shimizu [39] proves that warp(K1 #K2 ) ≤ max{warp(K1 ), warp(K2 )} for knots K1 and K2 . Her argument also applies to the warping span of links, and so warp(L1 #L2 ) ≤ max warp(L1 ), warp(L2 ) ≤ warp(L1 ) + warp(L2 ) for any connected sum L1 #L2 of links L1 and L2 . Hence the warping span of a link is an alternating distance. 2 Shimizu proves that changing a crossing in a knot diagram D can alter warp(D) by at most one. The proof when D is instead a link diagram is identical. Consequently,
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warp(L) ≤ dalt(L)
(2.6)
for any link L. 3. Obstructions and lower bounds The invariants under consideration are defined as a minimum over all diagrams or as a minimum over all projections to some surface. Typically, invariants of this form are difficult to compute, and so it will be useful to have obstructions and computable lower bounds for as many of the invariants as possible. The first obstruction comes from the hyperbolic geometry of the link complement. Menasco [27] proves that a prime, non-split alternating link is either a torus link or a hyperbolic link. Adams et al. [3] and Adams [4] extend this result to almost-alternating and toroidally alternating knots. Proposition 3.1. (Adams et al. [3]) Let K be a prime knot. If dalt(K) = 1 or galt (K) = 1, then K is either a torus knot or a hyperbolic knot. Since a knot has alternating genus and Turaev genus zero if and only if it is alternating, Inequality (2.5) implies the following corollary. Corollary 3.2. Let K be a prime knot. If gT (K) = 1, then K is either a torus knot or a hyperbolic knot. Our computable lower bounds arise from either Khovanov homology [22] or knot Floer homology [33,34]. The Khovanov homology of a link L, denoted by Kh(L), is a bigraded Z-module with homological grading i and polynomial (or quantum) grading j. The diagonal grading δ is defined by δ = j − 2i, and when Kh(L) is decomposed over summands with respect to the δ-grading, we write Kh(L) = δ Kh δ (L). The width of Kh(L), denoted by w(Kh(L)), is defined as 1 w Kh(L) = max δ Kh δ (L) = 0 − min δ Kh δ (L) = 0 + 1. 2 The factor of 1/2 is included in the definition of width since if L has an odd number of components, all δ-gradings where Kh δ (L) = 0 are even, and if L has an even number of components, all δ-gradings where Kh δ (L) = 0 are odd. Champanerkar and Kofman [7] and independently Wehrli [45] show that there is a complex whose generators correspond to spanning trees of the checkerboard graph of a diagram D of L and whose homology is the Khovanov homology Kh(L) of L. Champanerkar, Kofman, and Stotlzfus [9] use a relationship between spanning trees of the checkerboard graph of D and certain graphs embedded in the Turaev surface F (D) to show w Kh(L) − 2 ≤ gT (L).
(3.1)
The relationship between the Turaev surface and Khovanov homology is further explained by Dasbach and Lowrance [13]. Let F denote the vector space with two elements. We consider the “hat version” HF K(K) of the knot Floer homology of a knot K with coefficients in F. The invariant HF K(K) is a bigraded F-vector space with Maslov (or homological) grading m and Alexander (or polynomial) grading s. The diagonal grading δ is defined as δ = s − m, and when HF K(K) is decomposed over summands with respect to the δ-grading, K(K), denoted by w(HF we write HF K(K) = δ HF K δ (K). The width of HF K(K)), is defined as w HF K(K) = max δ HF K δ (K) = 0 − min δ HF K δ (K) = 0 + 1.
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Fig. 5. Left: a positive crossing. Right: a negative crossing.
Ozsváth and Szabó [32] show that there is a complex whose generators correspond to spanning trees of K(K). The author [25] uses the the checkerboard graph of a diagram D of K and whose homology is HF Ozsváth–Szabó spanning tree complex to prove that w HF K(K) − 1 ≤ gT (K).
(3.2)
Rasmussen [35] uses Lee’s spectral sequence [24] to show that the Khovanov complex of a knot K gives rise to a concordance invariant s(K). Suppose that K+ and K− are two knots such that K+ can be transformed into K− by changing a single positive crossing to a negative crossing in some diagram (see Fig. 5). Then s(K− ) ≤ s(K+ ) ≤ s(K− ) + 2.
(3.3)
Let σ(K) denote the signature of a knot with sign convention chosen so that the signature of the positive trefoil is −2. Cochran and Lickorish [10] show that σ(K− ) − 2 ≤ σ(K+ ) ≤ σ(K− ).
(3.4)
Abe [1] uses the behavior of the s-invariant and signature under crossing changes to show s(K) + σ(K) ≤ 2 alt(K).
(3.5)
Using work on knot signature of Murasugi [30] and Thistlethwaite [42] and the spanning tree complexes of Champanerkar, Kofman and Wehrli, Dasbach and the author [12] show that s(K) + σ(K) ≤ 2gT (L).
(3.6)
4. Some families of links and their alternating distances In this section we examine the three families of links: iterated Whitehead doubles of the figure-eight knot, modified torus links, and the (3, q)-torus links. We use these three families to prove Theorems 1.1 and 1.2. 4.1. Iterated Whitehead doubles of the figure-eight knot Let P be a knot embedded in a genus one handlebody Y . For any knot K, identify a regular neighborhood of K with Y such that the generator of H1 (Y, Z) is identified with a longitude of K coming from a Seifert surface. The image of P is a knot S, called a satellite of K. The knot P is the pattern for S, and K is the companion. Define the positive t-twisted Whitehead double of a knot K, denoted by D+ (K, t), to be the satellite of K where the pattern knot P is the t-twisted positive clasp knot given in Fig. 6. We use iterated Whitehead doubles of the figure-eight knot to show that the Turaev genus and dealternating number dominate the alternation number of a link.
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Fig. 6. The satellite of the figure-eight K with pattern P and companion K is the positive t-twisted Whitehead double D+ (K, t) of K. Each box labeled t indicates t positive full twists.
Let W0 be the figure-eight knot 41 . For each positive integer n, define Wn = D+ (Wn−1 , 0), that is Wn is the positive n-th iterated untwisted Whitehead double of the figure-eight knot. Define the family F(Wn) by F(Wn ) = {Wn | n ≥ 0, n ∈ Z}. Hedden [15] computes the knot Floer homology of Wn . Proposition 4.1. (Hedden [15]) Let F denote the field with two elements, and let Fk(m) denote the vector space Fk in homological grading m. Then n ⎧ 2n m n ⎪ F ⎪ m=0 (1−m) ⎪ ⎪ n ⎪ ⎪ n 2n+1 m ⎨ F F (0) m=0 (−m) HF K ∗ (Wn , s) ∼ = n ⎪ ⎪ 2n m n ⎪ ⎪ F(−1−m) ⎪ m=0 ⎪ ⎩ 0
s = 1, s = 0, s = −1, otherwise,
and w(HF K(Wn )) = n + 1. The alternation number, Turaev genus, and alternating genus of Wn behave according to the following proposition. Proposition 4.2. For each positive integer n, alt(Wn ) = 1, gT (Wn ) ≥ n,
and
galt (Wn ) > 1. Proof. Let n be a positive integer. Changing one of the crossings of the clasp in any Whitehead double transforms the knot into an unknot, and thus the unknotting number u(Wn) is one. Since the unknot is alternating, the inequality alt(K) ≤ u(K) holds for every knot, and hence alt(Wn ) ≤ 1. The knot Wn is non-alternating since w(HF K(Wn )) > 1, and so alt(Wn ) = 1. Inequality (3.2) states that for any knot K, we have w(HF K(K)) −1 ≤ gT (K), and hence Proposition 4.1 implies that gT (Wn ) ≥ n.
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Since the genus of Wn is one, it follows that Wn is prime, and because Wn is a satellite knot, Proposition 3.1 implies that galt (Wn ) = 1. Since Wn is non-alternating, we may conclude that galt (Wn ) > 1. 2 4.2. Modified torus links The modified torus link T(p, q) is obtained by changing certain crossings of a standard diagram of the (p, q)-torus link T (p, q). We use the natural embedding of T (p, q) on a torus to show that T(p, q) is toroidally alternating for many choices of p and q. The behavior of the Rasmussen s-invariant and knot signature under crossing changes imply that the Turaev genus and alternation number of T(p, q) can be arbitrarily large. p denote Let Bp denote the p-stranded braid group, let Δp ∈ Bp denote the braid σ1 σ2 · · · σp−1 , and let Δ the braid p = Δ
p−1
(−1)i+1
σi
(−1)p
= σ1 σ2−1 σ3 · · · σp−1 .
i=1
Define T(p, q) to be the closure of the braid Δq−1 p Δp . The link T (p, q) can be obtained from the diagram p−1 q of the closure of Δp by changing 2 crossings where the closure of Δqp is a familiar diagram of the (p, q)-torus link T (p, q). Define the family F(T(p, q)) by F T(p, q) = T(p, q) p, q ≥ 3, p, q ∈ Z . Gordon, Litherland, and Murasugi [14] give the following recursive algorithm for computing the signature of the torus link T (p, q). Theorem 4.3. (Gordon et al. [14]) Suppose that p, q > 0. The following recurrence formulas hold. (1) Suppose that 2p < q. (a) If p is odd, then σ(T (p, q)) = σ(T (p, q − 2p)) − p2 + 1. (b) If p is even, then σ(T (p, q)) = σ(T (p, q − 2p)) − p2 . (2) σ(T (p, 2p)) = 1 − p2 . (3) Suppose that p ≤ q < 2p. (a) If p is odd, then σ(T (p, q)) = 1 − p2 − σ(T (p, 2p − q)). (b) If p is even, then σ(T (p, q)) = 2 − p2 − σ(T (p, 2p − q)). (4) σ(T (p, q)) = σ(T (q, p)), σ(T (p, 1)) = 0, σ(T (2, q)) = 1 − q. In order to estimate the bounds in Inequalities (3.5) and (3.6), we first estimate the signature and Rasmussen invariant for torus knots and the modified torus knots T(p, q). Proposition 4.4. Let p and q be relatively prime integers with p ≥ 3 and q ≥ 3. Then 1 1 −(p − 1)(p − 2) − pq ≤ σ T (p, q) ≤ (p − 1)(p − 2) − (p − 1)q, 2 2 1 1 −(p − 1)(p − 2) − pq ≤ σ T(p, q) ≤ (p − 1)2 − (p − 1)q, and 2 2 pq − 2p − q + 2 ≤ s T (p, q) ≤ pq − p − q + 1.
(4.1) (4.2) (4.3)
Proof. Let k be an integer relatively prime to p with 0 < k < p. Rudolph [36] shows that the signature of the closure of a positive braid is negative, and hence σ(T (p, k)) ≤ 0. The unknotting number u(K) of any
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knot K satisfies |σ(K)| ≤ 2u(K). Kronheimer and Mrowka [23] show that the unknotting number of T (p, k) is 12 (p − 1)(k − 1), and thus −(p − 1)(p − 2) ≤ σ T (p, k) ≤ 0.
(4.4)
Let q = np + r where n is a non-negative integer and 0 < r < p. Suppose that n = 0. Then (p − 1)(p − 2) − 12 (p − 1)q = (p − 1)(p − 2) − 12 (p − 1)r ≥ 0 because 12 r ≤ p − 2. Thus 1 1 −(p − 1)(p − 2) − pq = −(p − 1)(p − 2) − pr 2 2 < −(p − 1)(p − 2) ≤ σ T (p, q) ≤0 1 ≤ (p − 1)(p − 2) − (p − 1)q, 2 and Inequality (4.1) is proven when n = 0. Now suppose that n > 0. Repeated applications of Theorem 4.3 yields
σ T (p, np + r) =
⎧ σ(T (p, r)) − 12 n(p2 − 1) ⎪ ⎪ ⎪ ⎪ ⎨ σ(T (p, r)) − 1 np2 2 ⎪ −σ(T (p, p − r)) − ⎪ ⎪ ⎪ ⎩ −σ(T (p, p − r)) −
1 2 (n 1 2 (n
p odd, n even, p even, n even,
+ 1)(p2 − 1) p odd, n odd, + 1)p2 + 2
(4.5)
p even, n odd.
Since 0 < r < p and 0 < p − r < p, Inequality (4.4) implies that −(p − 1)(p − 2) ≤ σ(T (p, r)) ≤ 0 and −(p − 1)(p − 2) ≤ σ(T (p, p − r)) ≤ 0. Combining these inequalities with Eq. (4.5) yields the inequalities 1 1 −(p − 1)(p − 2) − np2 − 1 ≤ σ T (p, q) ≤ (p − 1)(q − 1) − (n + 1) p2 − 1 . 2 2 Since q = np + r, it follows that pq = np2 + rp > np2 + 2, and hence 1 1 −(p − 1)(p − 2) − pq < −(p − 1)(p − 2) − np2 − 1 ≤ σ T (p, q) . 2 2 Likewise, since q = np + r, it follows that (p − 1)q = np2 + rp − np − r ≤ np2 + p2 − n − 1 = (n + 1) p2 − 1 . Hence 1 1 σ T (p, q) ≤ (p − 1)(q − 1) − (n + 1) p2 − 1 ≤ (p − 1)(p − 2) − (p − 1)q, 2 2 and thus Inequality (4.1) is proven for all n. Since T(p, q) can be obtained from T (p, q) via p−1 2 crossing changes, Inequality (4.2) follows from Inequalities (3.4) and (4.1). Rasmussen computes the s-invariant for positive knots, and applying his formula to T (p, q) yields s(T (p, q)) = pq − p − q + 1. Inequality (4.3) follows from this fact and Inequality (3.3). 2
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Corollary 4.5. Let p be a fixed integer with p ≥ 3. For any positive integer n, there exists a q relatively prime to p such that alt(T(p, q)) ≥ n and gT (T(p, q)) ≥ n. Proof. Proposition 4.4 implies that if p and q are relatively prime with p ≥ 3 and q ≥ 3 then 1 pq − 2p − q + 2 − (p − 1)(p − 2) ≤ s T(p, q) + σ T(p, q) 2 If p is fixed and q goes to infinity, then the left hand side of the previous inequality is eventually positive and grows without bound. Therefore for sufficiently large values of q (with p and q relatively prime), we have 1 pq − 2p − q + 2 − (p − 1)(p − 2) ≤ s T(p, q) + σ T(p, q) . 2 Hence Inequalities (3.5) and (3.6) imply that alt(T(p, q)) and gT (T(p, q)) can be arbitrarily large. 2 An alternate proof that gT (T(p, q)) can be arbitrarily large uses Inequality (3.1) and a computation of the Khovanov width of T(p, q). While the alternation number and Turaev genus of T(p, q) grow without bound, the next proposition shows that many such knots are either alternating or toroidally alternating. Proposition 4.6. Let p be an even integer and let q be an odd integer with p ≥ 4, q ≥ 3, and q = 1 mod p. Then galt T(p, q) ≤ 1. Proof. Let Σ be a Heegaard torus in S 3 and suppose that T (p, q − 1) is embedded on Σ by going p times around the longitude and q − 1 times around the meridian. Cut Σ along a meridian to obtain a cylinder C1 . p is embedded on a cylinder C2 so that the incoming strands meet Suppose a standard planar diagram of Δ one boundary component and the outgoing strands meet the other. Identify each boundary component of C1 with a boundary component of C2 so that the resulting surface Σ is again a Heegaard torus in S 3 with a diagram of T(p, q) projected onto it (via some projection π). Before we glue in the cylinder C2 each component of Σ − T (p, q − 1) is an annulus, and since q = p mod 1, p , the the number of components of Σ − T (p, q − 1) is strictly less than p. By surgering in the diagram of Δ annuli are separated into a disjoint union of disks. p by 1, 2, . . . , p from left to right. The incoming strands Label the incoming and outgoing strands of Δ labeled with odd numbers encounter an under-crossing first, while the incoming strands labeled with even integers encounter an over-crossing first. For the outgoing strands, the situation is reversed. Outgoing strands labeled with odd numbers most recently encountered over-crossings, while outgoing strands labeled with even integers most recently encountered under-crossings. Since p is even and q − 1 is even, the permutation on the strands induced by Δq−1 sends strands labeled with odd numbers to strands labeled with odd p numbers and sends strands labeled with even numbers to strands labeled with even numbers. Therefore, the projection π(T(p, q)) is alternating on Σ. Since T(p, q) has an alternating projection to a Heegaard torus where the complement of the projection is a disjoint union of disks, it follows that galt (T(p, q)) ≤ 1. 2 Fig. 7 shows that T(4, 3) and T(4, 4k + 3) for non-negative k have alternating projections to Heegaard tori in S 3 whose complements are disjoint unions of disks. The examples in Fig. 7 demonstrate that there is a Heegaard surface Σ on which the knot has an alternating projection whose complement is a collection of disks such that Σ is not the Turaev surface of any diagram of the knot. Armond, Druivenga, and Kindred [5] show how to ensure such a surface is a Turaev surface using Heegaard diagrams.
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Fig. 7. After identifying the two components of the boundary of each cylinder to obtain tori, the diagram on the left is an alternating projection of T (4, 3) to the torus, and the diagram on the right is an alternating projection of T (4, 4k + 3) to the torus for each k ≥ 0.
4.3. The (3, q)-torus knots Let F(T (3, q)) be the family of (3, q)-torus links where q is any integer. The (3, q)-torus knots have arbitrarily large alternation number, Turaev genus, c(T (3, q)) − spanT (3,q) (t), and dealternating number, but have warping span 0 or 1/2. Kanenobu [18] computes the alternation numbers of the (3, q)-torus knots, up to an additive error of at most one. Proposition 4.7. (Kanenobu [18]) For any positive integer n, alt T (3, 4) = alt T (3, 5) = 1, alt T (3, 6n + 1) = alt T (3, 6n + 2) = 2n, alt T (3, 6n + 4) = alt T (3, 6n + 5) = 2n or 2n + 1. Using Inequality (3.1) and work of Stošić [40] and Turner [44], the author [26] computes the Turaev genus of the (3, q)-torus knots. Abe and Kishimoto [2] independently compute the Turaev genus of the (3, q)-torus knots and also compute their dealternating numbers. Proposition 4.8. (Abe and Kishimoto [2], Lowrance [26]) Let n be a non-negative integer, and let i = 1 or 2. Then gT T (3, 3n + i) = dalt T (3, 3n + i) = n. Combining work of Jones and Murasugi, we obtain the following result about the difference between the crossing number and the span of the Jones polynomial of T (3, q). Proposition 4.9. (Jones [17], Murasugi [31]) Suppose that q is relatively prime to 3 and q > 3. Then c T (3, q) − span VT (3,q) (t) = q − 1.
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Fig. 8. The 3-braids σ1 σ2 (left) and σ1−1 σ2−1 (right) with their edges labeled by weights.
Proof. Jones [17] gives the following formula for the Jones polynomial of all torus knots:
VT (p,q) (t) =
t(p−1)(q−1)/2 1 − tp+1 − tq+1 + tp+q . 2 1−t
For the (3, q)-torus knots, this formula yields VT (3,q) (t) = tq−1 + tq+1 − t2q . Hence span VT (3,q) (t) = q + 1. Murasugi [31] shows that the crossing number of T (3, q) where q > 3 is 2q. Therefore c T (3, q) − span VT (3,q) (t) = q − 1.
2
An example of Shimizu [39] is easily generalized to the following result. Proposition 4.10. If q is an integer with |q| > 2, then warp(T (3, q)) = 12 . Proof. Suppose q > 0. We write T (3, q) as the closure of (σ1 σ2 )q , and T (3, −q) as the closure of (σ1−1 σ2−1 )q . Fig. 8 depicts the 3-braids σ1 σ2 and σ1−1 σ2−1 with edges labeled by weights 0, 1, and 2. Since the incoming weights are the same as the outgoing weights, these braids can be stacked q times to obtain diagrams of T (3, q) and T (3, −q) where the only weights are 0, 1, and 2. Hence warp(T (3, q)) = warp(T (3, −q)) ≤ 12 . Since T (3, q) is alternating if and only if |q| ≤ 2, the result follows. 2
4.4. Proofs of Theorems 1.1 and 1.2 The proofs of Theorems 1.1 and 1.2 are pieced together from the previous three subsections. Proof of Theorem 1.1. Proposition 4.2 shows that gT (Wn ) −alt(Wn ) ≥ n −1 and Inequality (2.4) shows that dalt(Wn ) − alt(Wn ) ≥ n − 1. Thus alt(F(Wn )) gT (F(Wn )) and alt(F(Wn )) dalt(F(Wn )). Inequality (2.3) implies that c(L) − span VL (t) dominates the alternation number on F(Wn ). Fix an even integer p ≥ 4 and let q range over all positive integers such that p and q are relatively prime, and q = 1 mod p. Proposition 4.6 states that galt (T(p, q)) = 1 while Corollary 4.5 implies that alt(T(p, q)) and gT (T(p, q)) go to infinity as q goes to infinity. Hence galt (F(T(p, q))) alt(F(T(p, q))) and galt (F(T(p, q))) gT (F(T(p, q))). Both Inequality (2.1) and Inequality (2.4) imply that galt (F(T(p, q))) dalt(F(T(p, q))). Inequality (2.3) implies that the difference c(L) − span VL (t) dominates the alternating genus on F(T(p, q)).
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Fig. 9. The crossing to the left is extended by the alternating tangle to the right.
Now let q be a positive integer not divisible by 3 with q > 3. Propositions 4.7, 4.8, 4.9, and 4.10 yield c T (3, q) − span VT (3,q) (t) = q − 1, dalt T (3, q) = q/3, gT T (3, q) = q/3, alt T (3, q) = q/3 or q/3 − 1, warp T (3, q) = 1/2,
and
which implies statements (3) and (4). 2 Proof of Theorem 1.2. Proposition 4.2 implies that for the positive n-th iterated untwisted Whitehead double Wn of the figure-eight knot, alt(Wn ) < galt (Wn ) for each positive integer n. Furthermore, we saw in the proof of Theorem 1.1 that galt (F(T(p, q))) alt(F(T(p, q))). Therefore the alternation number and alternating genus of a link are not comparable. 2 5. Further questions Abe and Kishimoto [2] showed that gT (D) ≤ dalt(D) for any link diagram D. For many diagrams, this inequality is strict, however the following question remains open. Question 1. Is there a link L such that gT (L) < dalt(L)? We now describe a method to construct diagrams where dalt(D) − gT (D) is arbitrarily large. Let x be a crossing in a link diagram D considered as a 2-tangle. An alternating 2-tangle τ is said to extend the crossing x, if there is some choice of resolutions of all but one crossing of τ such that the resulting tangle is isotopic to x through a planar isotopy fixing the endpoints of the tangle. If D is a diagram with crossing x that is extended by the rational tangle τ , then let D(x, τ ) be the diagram obtained by replacing the 2-tangle x with τ (see Fig. 9). Let D be a link diagram with c(D) crossings such that dalt(D) = k. Suppose that x is one of the k crossings changed to make D alternating, and suppose that τ is an alternating tangle with c(τ ) crossings extending x. In order to minimally transform D(x, τ ) into an alternating diagram, one must either change the k − 1 other crossing from above along with every crossing of τ or one must change every crossing other than the k crossings from above. Hence dalt D(x, τ ) = min c(D) − k, c(τ ) + k − 1 . One can show that |sA (D)| + |sB (D)| − c(D) = |sA (D(x, τ ))| + |sB (D(x, τ ))| − c(D(x, τ )), and therefore gT (D(x, τ )) = gT (D). Hence a suitably chosen sequence of alternating tangle extensions can force the gap
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between the dealternating number of a diagram and genus of the Turaev surface of the diagram to grow without bound. Question 1 being open means that it is not yet determined whether the dealternating number and the Turaev genus are actually different invariants. It is natural to ask whether Inequality (2.3) has an analog for the dealternating number. Question 2. Is dalt(L) ≤ c(L) − span VL (t) for any link L. Turaev [43] shows that gT (D) ≤ c(D) − span VL (t) for any diagram D of a link L. However, this approach does not work for the dealternating number because there exists a diagram where dalt(D) > c(D) − span VL (t). Let D be the usual diagram (5, −3, 2) pretzel knot, and let P (5, −3, 2) be the knot with diagram D. Then P (5, −3, 2) is knot 10125 in the Rolfsen table. We have that dalt(D) = 3 while c(D) = 10 and span VP (5,−3,2) (t) = 8. Thus dalt(D) = 3 > 2 = c(D) − span VP (5,−3,2) (t). Kim and Lee [21] prove that every non-alternating pretzel link has dealternating number one, and hence P (5, −3, 2) has a some diagram (not D) implying that dalt(P (5, −3, 2)) = 1. By examining the behavior of Whitehead doubles, we saw that the Turaev genus dominates the alternation number of a link. However, the following question remains open. Question 3. Are the alternation number and the Turaev genus of a link comparable? If the alternation number and Turaev genus of a link are comparable, then we must have alt(L) ≤ gT (L) for any link L. If they are not comparable, then there exists a link L such that gT (L) < alt(L), and hence Inequality (2.1) would imply an affirmative answer to Question 1. Inequality (2.6) follows from the fact that warp(D) ≤ dalt(D) for every link diagram D. Since Turaev genus dominates the warping span, the following question is natural. Question 4. Are the warping span and the Turaev genus of a link comparable? If warping span and Turaev genus are comparable, then warp(L) ≤ gT (L) for any link L. However, this cannot be proved in the same way as Inequality (2.6) because there exist diagrams D with warp(D) > gT (D). For example, let D be the diagram of the unknot obtained by taking the closure of the 2-braid σ1−1 σ1 σ1−1 σ1 σ1−1 . A straightforward computation shows that warp(D) = 2 while gT (D) = 1. If the warping span and Turaev genus are not comparable, then there exists a link L with gT (L) < warp(L). In order to confirm such an example, it would be useful to find lower bounds for the warping span of a link. The case of alternating genus is similar. Proposition 3.1 gives an obstruction for certain links to have alternating genus one, but beyond that there is no useful lower bound for the alternating genus of a link. This leads naturally to our final question. Question 5. Do links with arbitrarily large alternating genus or warping span exist? Acknowledgements The author is grateful for comments from Oliver Dasbach, Radmila Sazdanović, and Alexander Zupan. References [1] Tetsuya Abe, An estimation of the alternation number of a torus knot, J. Knot Theory Ramif. 18 (3) (2009) 363–379. [2] Tetsuya Abe, Kengo Kishimoto, The dealternating number and the alternation number of a closed 3-braid, J. Knot Theory Ramif. 19 (9) (2010) 1157–1181.
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